Seminormal λ-generated ideals on P _κ λ

Extract

In this paper we consider the problem of lifting properties of the Fréchet ideal I_κ = {X ⊆ κ: ∣X∣ < κ} on a regular uncountable cardinal κ, to an analogue about I_κλ , the ideal of not unbounded subsets of P_κλ. With this in mind, in §1 we introduce and study the class of seminormal λ-generated ideals on P_κλ. We shall see that ideals belonging to this class display properties which are clearly analogous to those of the Fréchet ideal on κ (for instance, with regard to saturation, normality and weak selectivity) and yet are closely related to I_κλ . Our results here show that if λ^<λ = λ, then many restrictions of I_κλ are weakly selective, nowhere precipitous and, quite suprisingly, seminormal (but nowhere normal). These latter two results suggest the question of whether any restriction of I_κλ can ever be normal. In §2 we prove that if κ is strongly inaccessible, λ^<κ = 2 ^λ and NS _κλ , the ideal of nonstationary subsets of P_κλ, has a mild selective property, then NS _κλ ∣A = I_κλ ∣A for some stationary A ⊆ P_κλ.

In [1] Baumgartner showed that if κ is weakly compact and P is the collection of indescribable subsets of κ, then P → (P, κ)². As a P_κλ analogue of indescribability, Carr (see [3]–[5]) introduced the λ-Shelah property, but was unable to derive the natural P_κλ analogue of Baumgartner's result, (where NSh _κλ is the normal ideal on P_κλ induced by the λ-Shelah property). In §3 we show that the problem lies in the fact that, as far as we know, NSh _κλ is not sufficiently distributive, and derive conditions which are sufficient and, in a sense, necessary to yield partitions related to .